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Already from this basic principles it can be concluded that intuitionism differs from Platonism and formalism, because neither does it assume a mathematical reality outside of us, nor does it hold that mathematics is a play with symbols according to certain fixed rules.
"Conway is the rare sort of mathematician whose ability to connect his pet mathematical interests makes one wonder if he isn't, at some level, shaping mathematical reality and not just exploring it," James Propp, a professor of mathematics at the University of Massachusetts Lowell, once told me.
The mathematical reality is that wealth is becoming more important, relative to income.
"We understand the mathematical reality of competing against an opponent with unlimited financial resources," he said.
Those who tout the virtues of the Electoral College should confront the mathematical reality of the inequities they are defending.
The notion that having fewer rooms means having more space clings stubbornly in the face of mathematical reality.
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Sanders prompted one or two boos from the emotional crowd of his supporters when he spelled out the mathematical realities to them once and for all, but left no doubt where he now stood.
THERE are certain mathematical realities associated with New York City private schools: There are more students than seats at the top-tier schools, at least three sets of twins will be vying head to head for spots in any class, and already-expensive tuition can only go up.
Even mathematical realities belong in the natural world: numbers at least natural ones as abstractions from clusters, and geometrical objects as abstractions from space.
They claim that while some textbooks which were restructured according to the calculus reform movement emphasized the use of technology, mathematical realities are ignored (Chappell & Kilpatrick, 2003).
It's the result of more than 100 years of ignoring the mathematical realities about the ways in which economics works because the methods we were using appeared to work for other reasons entirely.
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